Modelling space based methods

Partition-based methods address dimensionality by selecting input variables that are most relevant to efficient empirical modeling. The input space is partitioned by hyperplanes that are perpendicular to at least one of the input axes, as depicted in Fig. 6d. 

For optimization problems that are derived from (ordinary or partial) differential equation models, a number of advanced optimization strategies can be applied. Most of these problems are posed as NLPs, although recent work has also extended these models to MINLPs and global optimization formulations. For the optimization of profiles in time and space, indirect methods can be applied based on the optimality conditions of the infinite-dimensional problem using, for instance, the calculus of variations. However, these methods become difficult to apply if inequality constraints and discrete decisions are part of the optimization problem. Instead, current methods are based on NLP and MINLP formulations and can be divided into two classes ... 

Recent developments and prospects of these methods have been discussed in a chapter by Schneider et al. (2001). It was underlined that these methods are widely applied for the characterization of crystalline materials (phase identification, quantitative analysis, determination of structure imperfections, crystal structure determination and analysis of 3D microstructural properties). Phase identification was traditionally based on a comparison of observed data with interplanar spacings and relative intensities (d and T) listed for crystalline materials. More recent search-match procedures, based on digitized patterns, and Powder Diffraction File (International Centre for Diffraction Data, USA.) containing powder data for hundreds of thousands substances may result in a fast efficient qualitative analysis. The determination of the amounts of different phases present in a multi-component sample (quantitative analysis) is based on the so-called Rietveld method. Procedures for pattern indexing, structure solution and refinement of structure model are based on the same method. 

Two supervised methods are examined in this chapter, KNN and SIMCA. The KNN models are constructed using the physical closeness of samples in space. It is a simple method that does not rely on many assumptions about the data. SIMCA models are based on more assumptions and define classes using the position and shape of the object formed by the samples in row space. A multidimensional box is constructed for each class using PCA and tlie classification of future samples is perfonned by determining within which box the sample belongs. 

A different approach to mention here because it has some similarity to QM/MM is called RISM-SCF [5], It is based on a QM description of the solute, and makes use of some expressions of the integral equation of liquids (a physical approach that for reasons of space we cannot present here) to obtain in a simpler way the information encoded in the solvent distribution function used by MM and QM/MM methods. Both RISM-SCF and QM/MM use this information to define an effective Hamiltonian for the solute and both proceed step by step in improving the description of the solute electronic distribution and solvent distribution function, which in both methods are two coupled quantities. There is in this book a contribution by Sato dedicated to RISM-SCF to which the reader is referred. Sato also includes a mention of the 3D-RISM approach [6] which introduces important features in the physics of the model. In fact the simulation-based methods we have thus far mentioned use a spherically averaged radial distribution function, p(r) instead of a full position dependent function p(r) expression. For molecules of irregular shape and with groups of different polarity on the molecular periphery the examination of the averaged p(r) may lead to erroneous conclusions which have to be corrected in some way [7], The 3D version we have mentioned partly eliminates these artifacts. 

The kinetic properties of strontium sorption on clinoptilolite from both model solutions and natural seawater have been investigated in detail [252, 253]. These properties are determined by diffusion in both the zeolite microcrystals and the intercrystalline porous space [252]. Methods for determining the characteristic size of microcrystals by an ion-exchange technique are given in [253]. Such estimates provided a determination of their size-based contribution to the diffusion process. 

And model-based methods which are composed of quantitative model-based methods (such as analytical redundancy (Chow and Willsky, 1984), parity space (Gertler and Singer, 1990), state estimation (Willsky, 1976), or fault detection filter (Franck, 1990)) and qualitative model-based methods (such as causal methods digraphs (Shih and Lee, 1995), or fault tree (Venkatasubramanian, et ah, 2003)). 

FIGURE 2.2 Exploration of the chemical space includes the classification and analysis of databases. To generate structural models several methodologies have been developed, they can be classified into ligand-based and target-based methods. Models derived from these methods are frequently used as a source for virtual screening. 

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